Monetary Policy and the

Finance and Economics Discussion Series

Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C.

Monetary Policy and the Predictability of Nominal Exchange Rates

Martin Eichenbaum, Benjamin K. Johannsen, and Sergio Rebelo

2017-037

Please cite this paper as:

Eichenbaum, Martin, Benjamin K. Johannsen, and Sergio Rebelo (2017). “Monetary Pol- icy and the Predictability of Nominal Exchange Rates,” Finance and Economics Discus- sion Series 2017-037. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.037r1.

NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Monetary Policy and the

Predictability of Nominal Exchange Rates

Martin Eichenbaum Benjamin K. Johannsen

Sergio Rebelo October 2017

Abstract

This paper documents two facts about countries with floating exchange rates where monetary policy controls inflation using a short-term interest rate. First, the current real exchange rate predicts future changes in the nominal exchange rate at horizons greater than two years both in sample and out of sample. This predictability improves with the length of the horizon. Second, the real exchange rate is virtually uncorrelated with future inflation rates both in the short run and in the long run. We show that a large class of open-economy models is consistent with these findings and that, empirically and theoretically, the ability of the real exchange rate to forecast changes in the nominal exchange rate depends critically on the nature of the monetary regime.

∗The views expressed here are those of the authors and do not necessarily reflect the views of the Board of Governors, the Federal Open Market Committee, or anyone else associated with the Federal Reserve System. We thank Adrien Auclert, Charles Engel, Gaetano Gaballo, Zvi Hercovitz, Oleg Itskhoki, Dmitry Mukhin, Paulo Rodrigues, Christopher Sims, and Oreste Tristani for their comments and Martin Bodenstein for helpful discussions.

1 Introduction

This paper studies how the monetary policy regime affects the relative importance of nominal exchange rates (N ERs) and inflation rates in shaping the response of real exchange rates (RERs) to shocks. To describe our findings, we define the RER as the price of the foreign-consumption basket in units of the home-consumption basket and theN ER as the price of the foreign currency in units of the home currency.

We begin by documenting two facts about real and nominal exchange rates for a set of benchmark countries. These countries have two characteristics in common over our sample period: they have flexible exchange rates and the central bank uses short-term interest rates to keep inflation near its target level. Our first fact, is that the currentRER is highly negatively correlated with future changes in theN ER at horizons greater than two years. This correlation is stronger the longer is the horizon. Our second fact, is that theRER is virtually uncorrelated with future inflation rates at all horizons.

Taken together, these facts imply that the RER adjusts in the medium and long runs over- whelmingly through changes in theN ER, not through differential inflation rates. When a country’s consumption basket is relatively expensive, itsN EReventually depreciates by enough to move the RER back to its long-run level. These conclusions are consistent with those of Cheung, Lai, and Bergman (2004).¹

Critically, we argue that these facts depend on the monetary policy regime in effect. To show this dependency, we re-do our analysis for China which is on a quasi-fixed exchange rate regime versus the U.S. dollar; for Hong Kong which has a fixed exchange rate versus the U.S. dollar; and for the euro-area countries, which have fixed exchange rates with each other. In all of these cases, the currentRERis highly negatively correlated with future relative inflation rates. In contrast to the flexible exchange rate countries, theRERadjusts overwhelmingly through predictable inflation differentials.

Additional evidence on the importance of the monetary policy regime comes from a set of coun- tries that had crawling pegs or heavily managed floating exchange rates and then moved to floating exchange rates and inflation targeting. This set of countries consists of Brazil, Chile, Colombia, Indonesia, Israel, Mexico, South Korea, and Thailand. We show that when these countries adopted floating exchange rates and inflation targeting, the dynamic co-movements of theN ER, the RER, and inflation became qualitatively similar to those in our benchmark countries. This type of sen- sitivity to the monetary policy regime is precisely what we would expect given the Lucas (1976) critique.

Before discussing a class of models that accounts for our findings, we confront the concern that these findings might be spurious in the sense that they might primarily reflect small sample sizes and persistent RERs.² We address these concerns in two ways. First, using a bootstrap methodology,

1These authors use an alternative statistical methodology to study the behavior of the exchange rates for four Eu- ropean countries and Japan. Their sample spans the period between the collapse of the Bretton Woods system and the establishment of the euro.

2Similar concerns lie at the heart of ongoing debates about the predictability of the equity premium based on variables

we find that it is implausible that our empirical findings could be produced by a data generating process (DGP) with a very persistentRER that is uncorrelated with future changes in theN ER.

Second, we show that out-of-sample forecasts of theN ERbased on theRER beat a random walk forecast at medium and long horizons. We argue that this finding is extremely unlikely if theN ER is not predictable, regardless of whether the underlying DGP for the RER is stationary. Viewed overall, these results are strongly supportive of the view that our key empirical findings for the benchmark countries are not spurious.

Having established our key facts, we turn to the underlying economics. We show that there is a wide class of models consistent with the fact that, for our benchmark countries, the currentRER predicts future movements in the N ER. This consistency holds in models both with and without nominal rigidities. The key elements of these models are that monetary policy is governed by a Taylor rule and there is home bias in consumption.

We analyze versions of the same class of models in which the foreign central bank follows a managed float. We show that these models are consistent with the fact that, under a managed float, the RER is useful for predicting future movements in differential inflation rates. While the previous findings hold for all versions of the model that we consider, a dynamic stochastic general equilibrium (DSGE) model with nominal rigidities does the best job quantitatively.

We begin our theoretical analysis with a simple flexible-price model where labor is the only factor in the production of intermediate goods. The intuition for why this simple model accounts for our empirical findings about Taylor-rule regimes is as follows. Consider a persistent fall in domestic productivity or an increase in domestic government spending. Both shocks lead to a rise in the real cost of producing home goods that dissipates smoothly over time. Home bias means that domestically produced goods have a high weight in the domestic consumer basket. So, after the shock, the price of the foreign consumption basket in units of the home consumption basket falls, i.e. theRERfalls. The Taylor rule followed by both central banks keeps inflation relatively stable in the two countries. As a consequence, most of the adjustment in theRERoccurs through changes in the N ER. In our model, the N ER behaves in a way that is reminiscent of the overshooting phenomenon emphasized by Dornbusch (1976). After a technology shock, the foreign currency depreciates on impact and then slowly appreciates to a level consistent with the return of theRER to its steady-state value. The longer the horizon, the higher is the cumulative appreciation of the foreign currency. So in this simple model, the current RER is highly negatively correlated with the value of theN ER at future horizons, and this correlation is stronger the longer is the horizon.

These predictable movements in theN ER can occur in equilibrium because they are offset by the interest rate differential, i.e., uncovered interest parity (UIP) holds.

An obvious shortcoming of the flexible-price model is that purchasing power parity (PPP) holds at every point in time. To remedy this shortcoming, we modify the model so that monopolist producers set the nominal prices of domestic and exported goods in the local currency where they are sold. They do so subject to Calvo-style pricing frictions. For simplicity, suppose for now that there is a complete set of domestic and international asset markets. Consider a persistent fall in like the price–dividend ratio (see Stambaugh (1999); Boudoukh, Richardson and Whitelaw (2006); and Cochrane (2008)).

domestic productivity or an increase in domestic government spending. Both shocks lead to a rise in domestic marginal cost. The domestic firms that can reoptimize their prices increase them at home and abroad, so inflation rises. Because of home bias, domestic inflation rises by more than foreign inflation. The Taylor principle implies that the domestic real interest rate rises by more than the foreign real interest rate. So, domestic consumption falls by more than foreign consumption.

With complete asset markets, the RER is proportional to the ratio of foreign to domestic marginal utilities of consumption. So, the fall in the ratio of domestic to foreign consumption implies a fall in theRER. As in the flexible price model, the Taylor rule keeps inflation relatively low in both countries so that most of the adjustment in theRERis attributable to movements in theN ER. Again, the implied predictable movements in theN ERcan occur in equilibrium because they are offset by the interest rate differential, i.e. UIP holds.

While the intuition is less straightforward, our results are not substantively affected if we replace complete markets with incomplete markets or assume producer-currency pricing instead of local- currency pricing. Risk premiums aside, UIP holds conditional on the realization of many types of shocks to the model economy. We introduce shocks to the demand for bonds, for which UIP does not hold. So, when the variance of these shocks is sufficiently large, traditional tests of UIP applied to data from our model would reject that hypothesis.

Finally, we assess whether empirically-plausible versions of our model can quantitatively account for the facts that we document by studying an open-economy medium-sized DSGE version of our model. Among other features, the model allows for Calvo-style nominal wage and price frictions and habit formation in consumption of the type considered in Christiano, Eichenbaum, and Evans (2005).

A key question is whether the models we study are consistent with other features of the data stressed in the open-economy literature. It is well known that, under flexible exchange rates, real and nominal exchange rates co-move closely in the short run (Mussa (1986)). This property, and the fact that RERs are highly inertial (Rogoff (1996)) constitute bedrock observations that any plausible open-economy model must be consistent with. We show that our medium-sized DSGE model with nominal rigidities is, in fact, consistent with these observations.

Finally, we show that our DSGE model can quantitatively account for the extent to whichRER- based medium- and long-run forecasts of theN ERoutperform random walk forecasts. Specifically, the model is consistent with the fact that the comparative advantage of RER-based forecasts increases with the forecasting horizon. In addition, the model accounts quantitatively for the average ratio of the root mean squared prediction error (RMSPE) of theRER-based and random walk-based forecasts at all horizons.

Our work is related to four important strands of literature. The first strand demonstrates the existence of long-run predictability in N ERs (e.g. Mark (1995) and Engel, Mark, and West (2007)). Our contribution here is to show that the ability of the RER to predict the N ER at medium and long-run horizons depends critically on the monetary policy regime in effect. The second strand of the literature, which goes back to Meese and Rogoff (1983), studies the out-of- sample predictability of theN ER. Authors like Engel and West (2004, 2005) and Molodtsova and

Papell (2009) have proposed using variables that might enter into a Taylor rule to improve out- of-sample forecasting. Such variables includes output gaps, inflation, and possibly RERs. Rossi (2013) provides a thorough review of this literature. Recently, Cheung, Chinn, Pascual, and Zhang (2017) highlight the potential role of theRERin helping forecast theN ER. Ca’Zorzi, Muck, and Rubaszek (2016) study the forecasting performance of the Justiniani and Preston (2010) DSGE model. Citing an earlier version of this paper, these authors note the potential usefulness of the RER in forecasting the N ER. Our contribution relative to these two papers is to thoroughly document that role and show how it depends on the monetary policy regime.

The third strand of the literature seeks to explain the persistence of RERs. See, for example, Rogoff (1996); Kollmann (2001); Benigno (2004); Engel, Mark, and West (2007); and Steinsson (2008). Our contribution relative to that literature is to show that we can account for the relation- ship between the RER and future changes in inflation and the N ER in a way that is consistent with the observed inertia in theRER.

The fourth strand of the literature emphasizes the importance of the monetary regime for the behavior of theRER. See, for example, Baxter and Stockman (1989); Henderson and McKibbin (1993); Engel, Mark, and West (2007); and Engel (2012). Our contribution relative to this literature is to document the importance of the monetary regime in determining the relative roles of inflation and theN ER in the adjustment of theRER to its long-run levels.

Our paper is organized as follows. Section 2 contains our empirical results. Section 3 describes a sequence of models consistent with these results. We start with a model that has flexible prices and complete asset markets and where labor is the only factor in the production of intermediate goods.

We then replace complete markets with a version of incomplete markets where only one-period bonds can be traded. Next, we introduce Calvo-style frictions in price setting. In Section 4, we consider an estimated medium-scale DSGE model. Section 5 concludes.

2 Some empirical properties of exchange rates

In this section, we present our empirical results regarding N ERs, RERs, and relative inflation rates. We use consumer price indexes for all items and average quarterly N ERs versus the U.S.

dollar.

2.1 Data

We initially focus on a benchmark group of advanced economies—Australia, Canada, Norway, Sweden, and Switzerland—that had floating exchange rates in the period from 1973 to 2007.³ In choosing the sample period, we face the following trade off. On the one hand, we would like as long a time series as possible. On the other hand, we would like the monetary regime to be reasonably stable in our sample. To balance these considerations, we exclude from our sample data from 2008 to the present because short-term nominal interest rates in the United States were at or near

3Unless indicated otherwise, a year means that the entire year’s worth of data was used.

their effective lower bound. We include data since 2008 as a part of our robustness analysis. We exclude Japan from our set of benchmark countries because its short-term interest rates have been at or close to the effective lower bound since 1995. We exclude the United Kingdom, which left the European Exchange Rate Mechanism of the European Monetary System in 1992 after a large devaluation. We include data from both Japan and the United Kingdom in our robustness analysis, where we also consider countries that eventually adopted the euro.⁴

We compare results for the benchmark flexible exchange rate economies with those for China (from 1994 through 2007), which has been on a quasi-fixed exchange rate vis-`a-vis the U.S. dollar, and for Hong Kong (from 1985 through 2007), which has a fixed exchange rate vis-`a-vis the U.S.

dollar. We also analyze data starting in 1999 for France, Ireland, Italy, Portugal, and Spain where the RER and relative inflation rates are defined relative to Germany. In addition, we consider a group of countries that had crawling pegs or heavily managed floating exchange rates and then moved to floating exchange rate regimes along with a form of inflation targeting. This set of countries consists of Brazil, Chile, Colombia, Indonesia, Israel, Mexico, South Korea, Thailand, and Turkey.

2.2 Results for flexible exchange rate countries

We define theRER for countryirelative to the United States as:

RER_i,t = N ER_i,tP_i,t

P_t , (1)

where N ERi,t is the nominal exchange rate, defined as U.S. dollars per unit of foreign currency.

The variablesP_tandP_i,t denote the consumer price index in the U.S. and in countryi, respectively.

We assume that the RER is stationary and offer supporting evidence later in this section. Given this assumption, theRER must adjust back to its mean after a shock via changes in theN ER or changes in relative prices.

Figure 1 displays scatter plots for Canada of the log(RERi,t) against log (N ER_i,t+h/N ERi,t) at different horizons,h. The analogue figures for the other benchmark flexible exchange rate countries are displayed in the appendix. Two properties of this figure are worth noting. First, consistent with the notion that exchange rates behave like random walks at high frequencies, there is no obvious relationship between the log(RERi,t) and log (N ER_i,t+h/N ERi,t) at a one-year horizon.

However, as the horizon expands, the correlation between log (RER_i,t) and log (N ER_i,t+h/N ER_i,t) rises. The negative relation is very pronounced at longer horizons. This pattern holds for all of the benchmark flexible exchange rate countries included in the appendix.

4For bilateral exchange rate data between the United States and other countries, we use the H.10 exchange rate data published by the Federal Reserve, available at http://www.federalreserve.gov/releases/H10/Hist. We compute quarterly averages of the daily data. When the H.10 data do not include a country, we use exchange rate data from the International Monetary Fund’s International Financial Statistics database. For price indexes, we also use the International Monetary Fund’s International Financial Statistics database. When consumer price indexes are not available from the International Financial Statistics database, we use data from the Organization for Economic Cooperation and Development (OECD), which were downloaded from FRED, a database maintained by the Federal Reserve Bank of St. Luois (“Main Economic Indicators - complete database,” Main Economic Indicators (database)).

2.2.1 Nominal exchange rate regressions

We now discuss results based on the followingN ER regression:

log

N ER_i,t+h N ER_i,t

=α^{N ER}_i,h +β_i,h^{N ER}log(RER_i,t) +ε^{N ER}_i,t,t+h, (2) for country i at horizon h = 1,2, . . . , H years. Panel (a) of Table 1 reports estimates of β_i,h^{N ER}, along with standard errors, for the benchmark flexible exchange rate countries.⁵ A number of features are worth noting. First, for every country and every horizon, the estimated value ofβ_i,h^{N ER} is negative. Second, for almost all countries, the estimated value ofβ_i,h^{N ER} is statistically significant at three-year horizons or longer. Third, in most cases, the estimated value of β_i,h^{N ER} increases in absolute value with the horizon,h. Moreover,β_i,h^{N ER}is more precisely estimated for longer horizons.

Panel (a) of Table 1 also reports the R²s of the fitted regressions. Consistent with the visual impression from the scatter-plots, the R²s are relatively low at short horizons but rise with the horizon. Strikingly, for the longest horizons, the R² exceeds 50 percent for all of our benchmark countries and is 88 percent for Canada.

Taken together, the results in Table 1 strongly support the conclusion that, for our benchmark countries, the currentRERis strongly correlated with changes in futureN ERs, at horizons greater than roughly two years.

2.2.2 Relative price regressions

We now consider results based on the following relative-price regression:

log

P_i,t+h/P_t+h P_i,t/P_t

=α^π_i,h+β_i,h^π log(RER_i,t) +ε^π_i,t,t+h. (3) This regression quantifies how much of the adjustment in theRER occurs via changes in relative rates of inflation across countries. Panel (a) of Table 2 reports our estimates and standard errors for the slope coefficientβ_i,h^π . In most cases, the coefficient is statistically insignificant, though positive.

In some cases, it is negative instead of positive. Panel (a) of Table 2 also reports the R²s of the fitted regressions. These R²s are all much lower than those associated with regression (2). These results as a whole suggest that very little of the adjustment in the RER occurs via differential inflation rates. This conclusion is consistent with the results of Cheung, Lai, and Bergman (2004) based on an earlier sample period for Japan and four European countries.

2.2.3 Robustness: Other countries

We now assess the robustness of the previous results by considering other advanced economies with flexible exchange rates—the euro area, Japan, and the United Kingdom. Because the samples for these countries are relatively short, we only estimate regressions (2) and (3) out to a five-year

5We compute standard errors using a Newey-West estimator with the number of lags equal to the forecasting horizon plus two quarters.

horizon.⁶ Our results are reported in panel (b) of Table 1 and panel (b) of Table 2. The estimated regression coefficients are similar to those obtained for the benchmark countries. The appendix reports results for both these countries and the benchmark countries when we extend the sample to end in 2016:Q4. This change in sample period has little effect on our results.

2.3 Sensitivity to monetary policy

Our basic hypothesis is that the process by which the RER adjusts to shocks depends critically on the monetary policy regime. We provide two types of evidence in favor of this hypothesis.

First, we redo our analysis for countries that are on fixed or quasi-fixed exchange regimes. Second, we consider countries that, initially, heavily managed their exchange rates but later allowed their exchange rates to float.

2.3.1 Fixed and quasi-fixed exchange rates

In this subsection, we report the results of redoing our analysis for countries with fixed or quasi- fixed exchange rates. Results for China and Hong Kong, which have quasi-fixed and fixed exchange rates, respectively, are reported in panel (c) of Table 1 and panel (c) of Table 2. Several features of these results are worth noting. First, the estimated values of β_i,h^{N ER} are small relative to the estimates for our benchmark countries. Second, values ofβ_i,h^π are large relative to the estimates for our benchmark countries and statistically significant at every horizon. Third, the estimated value of β_i,h^π rises with the horizon, h. Fourth, the R² values associated with regression (3), reported in panel (c) of Table 2, are large and increase with the horizon.

We also consider several euro-area countries—France, Ireland, Italy, Portugal, and Spain—vis-

`

a-vis Germany. For these countries, the N ER is fixed. Results for regression (3) are reported in Table 3. As was the case for China and Hong Kong vis-`a-vis the United States, the estimated values ofβ_i,h^π are large, rise in magnitude with the horizon, and are statistically significant at long horizons. In addition, the R² values are large and increase with the horizon, with regression (3) explaining 94 percent of relative price movements between Germany and Portugal at a five-year horizon.

In sum, for economies with fixed or quasi-fixed exchange rates, theRERadjusts overwhelmingly through predictable inflation differentials, not through changes in theN ER.

2.3.2 Countries with changes in exchange rate policy

In this subsection, we redo our analysis for a set of countries—Brazil, Chile, Colombia, Indonesia, Israel, Mexico, South Korea, and Thailand—that had crawling pegs or heavily managed floating exchange rates and then adopted floating exchange rates.

6We begin the sample for the euro in 1999, when it was created. We start the sample for Japan in 1973 and end the sample in 1994, because Japan has had nominal interest rates near their effective lower bound since the mid-1990s.

We begin the sample for the United Kingdom in 1993 because the United Kingdom exited the European Exchange Rate Mechanism of the European Monetary System after a large devaluation in 1992.

We consider two sample periods. The first sample is from 1984:Q1 to 2016:Q4 and covers periods in which all of the countries moved from a managed exchange rate to a floating exchange rate. The second sample spans the period from 1999:Q1 to 2016:Q4. We include the period in which the zero lower bound (ZLB) is binding in the United States and in some other countries in order to have enough observations to estimate our regressions at a five-year horizon. Our experience with the benchmark countries suggests that including the ZLB period has a mild effect on the coefficients andR²s of regressions (2) and (3).

Tables (4) and (5) report our estimates ofβ^{N ER}_i,h andβ_i,h^π , as well as theR²s from the regressions.

In contrast to our benchmark countries, for the sample starting in 1984, the estimates ofβ_i,h^{N ER}and β_i,h^π , and the R² values, do not follow the consistent pattern observed for the benchmark flexible exchange rate countries. In addition, the estimates display no apparent pattern across the countries considered.

Tables (4) and (5) also report results for the sample starting in 1999. Notice that for every country except Turkey and every horizon, the estimates ofβ_i,h^{N ER} are negative, grow in magnitude with the horizon, and are statistically different from zero at longer horizons. In addition (again with the exception of Turkey), the R² values for regression (2) using the sample starting in 1999 are much larger than the analogousR²s from the full sample. By contrast, the estimates ofβ_i,h^π are relatively small in the sample starting in 1999, as are theR² values for regression (3).

Clearly, the post-1999 sample produces results that are more similar to those obtained with our benchmark flexible exchange rate countries. We view these results as being supportive of our hypothesis that the monetary policy regime is a central determinant of the way that the RER adjusts to shocks.

3 Are the empirical correlations spurious?

In the previous section, we argue that for our benchmark countries, changes in theN ER at long horizons display a strong negative correlation with the current level of the RER. A potential problem with this result is that if theRERis very persistent, we might statistically find in-sample predictability when none is actually present. Boudoukh et al. (2006) make this point in the context of the literature on equity returns predictability. In our context, their critique translates into the statement that the asymptotic standard errors for the regression coefficients reported in the previous section severely understate the importance of sampling uncertainty.

In this section, we address these concerns in two steps. Following the approach proposed by Boudoukh et al. (2006), we examine the small-sample properties of the Wald statistic for the test that the slope coefficients in regression (2), β_i,h^{N ER}, are zero at all horizons. Under the null hypothesis that the RER is a stationary process, we construct bootstrapp-values, which provide strong evidence against the hypothesis thatβ_i,h^{N ER} are all zero.

Analogue exercises conducted under the null hypothesis that theRERis a difference-stationary process turn out to have very low power, reflecting the diffuse nature of the small-sample distribution of the slope coefficients. Fortunately, in our case, tests based on out-of-sample forecasts of theN ER

are more powerful. We show that over medium- and long-run horizons, our forecasts of the N ER outperform random-walk forecasts. As discussed later, this finding is unlikely to reflect sampling uncertainty regardless of whether theRERhas a unit root.

3.1 Testing whether slope coefficients are zero

In this subsection, we test the joint null hypothesis that the slope coefficients in regression (2) are zero at all horizons up to 40 quarters (10 years), i.e.,

β_i,1^{N ER}=β^{N ER}_i,2 =· · ·=β_i,40^{N ER}= 0. (4) For each countryi, we jointly estimate the slope coefficients β_i,h^{N ER} and compute the Wald statis- tic under the null hypothesis (4).⁷ We focus attention on our benchmark flexible exchange rate countries so that we have enough data to include regressions with a horizon of 10 years.

Because theRERis highly persistent, we find in simulations that tests based on the asymptotic distribution of the Wald statistic have poor size. Accordingly, we test the null hypothesis (4) using the following bootstrap procedure. We assume that the stochastic processes forN ERi,tandRERi,t

are given by

log

N ERi,t

N ERi,t−1

= ε^{N ER}_i,t , (5)

Ai(L) log (RERi,t) = ε^RER_i,t . (6)

Here,A_i(L) is a polynomial in the lag operator with roots inside the unit circle so that the RER is a stationary process. The random variablesε^{N ER}_i,t and ε^RER_i,t are uncorrelated over time (though potentially correlated within a period). This DGP embeds the assumption that changes in the N ER are unpredictable at all horizons.⁸ We consider up to 10 lags in Ai(L) and choose the lag length separately for each country using the Akaike information criterion (AIC).⁹ Given the estimates of A_i(L), we back out a time series for ε^RER_i,t and ε^{N ER}_i,t from the observed data. We then jointly bootstrap ε^{N ER}_i,t and ε^RER_i,t to compute 10,000 synthetic time series, each of length equal to our actual sample period.¹⁰ For each synthetic time series, we estimate regression (2) for h= 1,2, . . . ,40 and compute the corresponding Wald statistics to produce a bootstrap distribution of that statistic.

Table 6 reports the fraction of the bootstrap Wald statistics that are larger than the correspond- ing Wald statistic that we computed in the data. With the exception of Norway, we can reject the null hypothesis (4) at the 5 percent significance level. For Norway, we can reject it at the 10 percent

7We compute standard errors using a Newey-West estimator with the number of lags equal to the forecasting horizon plus two quarters.

8Note that if log (N ERi,t/N ER_i,t−1) has a non-zero mean, that property is reflected in the fitted shocks from which we construct the bootstrap samples.

9The AIC selected four lags for Australia, seven lags for Canada, eight lags for Norway, four lags for Sweden, and four lags for Switzerland.

10We use 100 periods of initial burn in for our bootstraps.

significance level. Based on these tests, we infer that the negative correlations between theRER and the future changes of theN ER that we documented are unlikely to be spurious.

3.2 Out-of-sample forecasts

In this subsection, we use out-of-sample forecasting performance to test the null hypothesis that theN ER is not predictable. In practice, quarterly consumer price indexes are available with one period lag. To avoid any look-ahead bias, we measure the RER for country i using lagged price indexes so that

RERi,t≡ N ERi,tPi,t−1

Pt−1

. (7)

Our forecasting equation for theN ER is log

N ER_i,t+h N ERi,t

=α^{N ER}_i,h +β_h^{N ER}log(RER_i,t) +ε^{N ER}_i,t,t+h. (8) Notice that the parameter β_h^{N ER} is common across countries. This specification corresponds to a balanced panel with country-specific intercepts (α^{N ER}_i,h ) and common slopes.¹¹ We set the training period for the regression to the horizon of the forecast,h, plus 40 quarters.

We assess our ability to forecast the N ERrelative to a forecast of no change. The latter is the benchmark in the literature and corresponds to the assumption that the N ER is a random walk without drift. Define the RMSPE for countryiassociated with forecasts based on equation (8) as

σ_i,B,h=





 1 Ti,h

Ti,h

X

t=0

f_i,t,t+h−log

N ER_i,t+h N ERi,t

2





1/2

. (9)

Here, T_i,h denotes the number of forecasts for log(N ER_i,t+h/N ERi,t) in our sample, and f_i,t,t+h is the forecast of log(N ER_i,t+h/N ER_i,t) based on equation (8). We denote by σ_i,RW,h the corre- sponding RMSPE associated with the no-change forecast from a random walk model.

For each country i, we report the ratio of the RMSPE associated with the benchmark and random walk specifications, σi,B,h/σi,RW,h. We also compute a pooled RMSPE implied by our forecasting equation for all of the countries in our sample, defined as

σ_B,h =





 1 P

iT_i,h X

i Ti,h

X

t=0

f_i,t,t+h−log

N ER_i,t+h N ER_i,t

2





1/2

. (10)

We denote byσ_RW,h the pooled RMSPE implied by the random walk forecast and report the ratio of the pooled RMSPEs,σB,h/σRW,h.

We initially limit the analysis to our benchmark countries. Panel (a) of Table (7) reports relative RMSPEs for each country and for the pooled sample. Forecasts based on equation (8) outperform

11In adopting this approach, we follow Mark and Sul (2001), Groen (2005), and Engle, Mark, and West (2007), who use panel error-correction models to improve the forecasting power of exchange rate models.

the random walk model at horizons greater than two years. Remarkably, at the four- and seven- year horizons, forecasting equation (8) outperforms the random walk by 23 percent and 45 percent, respectively.¹²

We now formally test the hypothesis that the relative RMSPEs reported in panel (a) of Table 7 were generated by a DGP in which theN ERis a random walk. Under this hypothesis, changes in theN ER should not be predictable. We test this hypothesis using a bootstrap procedure similar to the one described in the previous subsection. In particular, we assume thatN ERi,t andRERi,t

are generated by equations (5) and (6), where we replaceRER_i,t with RER_i,t. The lag length of A_i(L) is chosen using the AIC.¹³ We construct 10,000 synthetic time series, each of length equal to the size of our sample, by randomly selecting a sequence of estimated disturbances. We jointly sample the disturbances so as to preserve contemporaneous correlations across theN ERandRER and across countries.¹⁴ For each synthetic time series, we compute forecasts based on equation (8) and the random walk without drift. Using these forecasts, we compute RMSPEs for each country and for the pooled countries.

Panel (b) of Table 7 shows the percentage of bootstrap simulations in which the value of the relative RMSPE is less than or equal to the analogue number reported in panel (a) at different horizons. The column labeled “Years 3–7” reports the percentage of bootstrap simulations where the relative RMSPEs are lower than in the data for all yearly horizons 3 through 7. For the horizon-specific tests usingσB,h/σRW,h, we can reject the random walk hypothesis at the 1 percent significance level using the one-quarter forecasts and at the 5 percent significance level for all individual horizons of at least three years. At the five-, six-, and seven-year horizons, we can reject the null hypothesis at the 1 percent significance level. For the joint test of yearly horizons 3–7, we can also reject the random walk hypothesis at the 1 percent significance level. There is some variability in the results for different countries and horizons. But the joint-horizon test provides very strong evidence against the random walk hypothesis for all of our benchmark countries.

Panel (c) of Table 7 reports robustness results forσ_B,h/σ_RW,h. The first row repeats our bench- mark results. The second row reports results for the case in which we use log(RER_i,t) instead of log(RERi,t) in forecasting equation (8). The results we obtain are very similar to the benchmark case. The third row reports the results of extending the sample period until the end of 2016.

There is a mild overall deterioration in forecasting performance at long horizons. The fourth row reports results obtained by adding Japan to our benchmark specification with the sample ending in December 2016. There is a further mild deterioration in forecasting performance at long horizons.

The fifth row reports results based on an unbalanced panel that includes the euro area starting in 1999:Q1, the United Kingdom starting in 1993:Q1, and Japan starting in 1973:Q1 and ending

12Additional recent evidence against random-walk-based forecasts for theN ER comes from Cheung, Chinn, Pascual, and Zhang (2017). These authors examine the ability of a host of economic models to forecastN ERs. They find that, relative to random walk forecasts, relative-purchasing-power-parity-based forecasts outperform other economic models.

13The AIC selected four lags for Australia, seven lags for Canada, eight lags for Norway, four lags for Sweden, and four lags for Switzerland.

14We again have a burn-in period of 100 quarters so that the initial values of log(RERi,t) are different across bootstrap samples.

in 1994:Q4.¹⁵ The results are about the same as the benchmark results at short horizons and only somewhat worse at long horizons. Still, the model outperforms the random walk for all horizons.

At the seven-year horizon, the RMSPE associated with our forecasting equation is 40 percent lower than that associated with a random walk.

The panel structure of our benchmark specification assumes that the slope coefficients are the same across all countries. A natural question is, how sensitive are our results to this assumption?

The sixth row of Table 7, labeled “Country-by-country regressions,” reports results obtained by estimating separate slope coefficients for each country. There is a slight deterioration in forecasting performance. But, even without imposing the panel structure, the model outperforms the random walk at long horizons (by 41 percent at the seven-year horizon).

To this point, we have maintained the assumption that the RER is stationary. To assess the robustness of our results, we redo the out-of-sample bootstrap exercises assuming that log(RER_i,t) is difference stationary. In particular, we assume that

Bi(L)(1−L) log RERi,t

=ε^RER_i,t . (11) Here,Bi(L) is a polynomial in the lag operator with roots inside the unit circle. We maintain the assumption that changes in theN ER are given by (5). As previously, we choose the lag length by the AIC and compute the relative RMSPEs.¹⁶ The implied p-values are reported in panel (d) of Table 7. The critical point is that the results we obtain are very similar to those reported in panel (b) of that table. We infer that our results are not sensitive to whether we assume that theRER has a unit root.¹⁷

In summary, the results reported in this section strongly support the view that changes in the N ER are predictable at medium- and long-run horizons. By implication, it is highly statistically unlikely that the correlations documented in the previous section are spurious.

4 Interpreting our empirical results: Economic models

In this section, we use a sequence of economic models to interpret the empirical findings documented earlier. We begin with a flexible-price, two-country, complete-markets model, allowing for different specifications of monetary policy, a Taylor rule, an exogenous money growth rule, and a regime where one country seeks to dampen fluctuations in theN ER.

Next, we consider a sticky-price model with an incomplete-markets setting in which the only assets traded internationally are bonds. It turns out that the complete- and incomplete-markets versions of our model have very similar implications.

Finally, we consider a medium-sized DSGE model with Calvo-style nominal price and wage

15The euro did not exist until 1999. The United Kingdom left the European Exchange Rate Mechanism in 1992 after Black Wednesday. Japan’s short-term interest rate has been at or near the ZLB since 1995.

16The AIC selected one lag for Australia, three lags for Canada, one lag for Norway, three lags for Sweden, and one lag for Switzerland.

17The results are not sensitive to assuming that theRERis a random walk.

rigidities in which producers set prices in local currencies. We allow for technology shocks in each country and shocks to the demand for domestic bonds. The latter shocks imply that unconditional UIP does not hold in our model.

4.1 Flexible-price, complete-markets model

The model consists of two completely symmetric countries. We first describe the households’

problems and then discuss the firms’ problems.

4.1.1 Households

The domestic economy is populated by a representative household whose preferences are given by Et

∞

X

j=0

β^j

"

log (Ct+j)− χ

1 +φL^1+φ_t+j +µ(M_t+j/P_t+j)^1−σM 1−σM

#

. (12)

Here,C_tdenotes consumption, L_thours worked,M_tend-of-period nominal money balances,P_t the price of consumption goods, and Et the expectations operator conditional on time-t information.

We assume that 0< β <1,σ_M >1, andχ and µare positive scalars.

Households can trade in a complete set of domestic and international contingent claims. The domestic household’s flow budget constraint is given by

B_H,t+N ER_tB_F,t+P_tC_t+M_t=Rt−1BH,t−1+N ER_tR_t−1^∗ BF,t−1+W_tL_t+T_t+Mt−1. (13) Here,BH,tandBF,tare nominal balances of home and foreign bonds;N ERtis the nominal exchange rate, defined as in our empirical section to be the price of the foreign currency unit (units of home currency per unit of foreign currency);R_tis the nominal interest rate on the home bond; R^∗_t is the nominal interest rate on the foreign bond; W_t is the nominal wage rate; and T_t denotes nominal lump-sum profits and taxes. For notational ease, we have suppressed the household’s purchases and payoffs of contingent claims. With complete markets, the presence of one-period nominal bonds is redundant since these bonds can be synthesized using state-contingent claims.

The first-order conditions with respect to labor supply, money balances, and consumption are χL^φ_tCt= W_t

Pt

, (14)

µ M_t

Pt

−σM

=

R_t−1 Rt

1 Ct

, (15)

1 =βRtEt

Ct

Ct+1πt+1

, (16)

where π_t ≡ P_t/Pt−1 denotes the inflation rate. Equation (15) characterizes money demand by domestic agents. Since households derive utility only from their country’s money, domestic agents do not hold foreign money balances.

We use stars to denote the prices and quantities in the foreign country. The preferences of the foreign household are given by

Et

∞

X

j=0

β^j





log C_t+j^∗

− χ

1 +φ L^∗_t+j1+φ

+µ

M_t+j^∗ /P_t+j^∗ 1−σ_M

1−σ_M





. (17)

The foreign household’s flow budget constraint is given by

B_F,t^∗ +N ER⁻¹_t B_H,t^∗ +P_t^∗C_t^∗+M_t^∗ =R^∗_t−1BF,t−1+N ER⁻¹_t Rt−1B_H,t−1^∗ +W_t^∗L^∗_t+T_t^∗+M_t−1^∗ . (18) The first-order conditions for the foreign household with respect to labor supply, money balances, and consumption are

χ(L^∗_t)^φC_t^∗ = W_t^∗

P_t^∗, (19)

µ M_t^∗

P_t^∗ −σM

=

R^∗_t−1 R^∗_t

1

C_t^∗, (20)

1 =βR^∗_tEt

C_t^∗

C_t+1^∗ π^∗_t+1. (21) As in our empirical section, we define the real exchange rate, RER_t, as the price of the foreign consumption good in units of the home consumption good:

RERt= N ERtP_t^∗

P_t . (22)

With this definition, an increase inRERt corresponds to a rise in the relative price of the foreign good.

Complete markets and symmetry of initial conditions imply C_t

C_t^∗ =RER_t. (23)

Combining equations (21) and (23) we obtain 1 =βR^∗_tE_t C_t

C_t+1π_t+1

N ER_t+1

N ER_t . (24)

Equations (16) and (23) imply

1 =βRtEt

C_t^∗ C_t+1^∗ π_t+1^∗

N ERt

N ER_t+1. (25)

4.1.2 Firms

The domestic final good,Y_t, is produced by combining domestic and foreign goods (Y_H,t and Y_F,t, respectively) according to the technology

Y_t=h

ω^1−ρ(Y_H,t)^ρ+ (1−ω)^1−ρ(Y_F,t)^ρi_ρ¹

. (26)

Here,ω >0 controls the importance of home bias in consumption. The parameter ρ≤1 controls the elasticity of substitution between home and foreign goods. Similarly, the foreign final good, Y_t^∗, is produced by combining domestic and foreign goods (Y_H,t^∗ and Y_F,t^∗ , respectively) according to the technology

Y_t^∗ =h

ω^1−ρ Y_F,t^∗ ρ

+ (1−ω)^1−ρ Y_H,t^∗ ρi_ρ¹

. (27)

The domestic goods used in the production of the domestic final good (Y_H,t) and in the pro- duction of the foreign final good (Y_H,t^∗ ) are produced according to the technologies

YH,t= Z 1

0

XH,t(j)^ν−1ν dj _ν−1^ν

and Y_H,t^∗ = Z 1

0

X_H,t^∗ (j)^ν−1ν dj _ν−1^ν

. (28)

Here, XH,t(j) and X_H,t^∗ (j) are domestic intermediate goods produced by monopolist j using the linear technology

X_H,t(j) +X_H,t^∗ (j) =A_tL_t(j) . (29) The variable L_t(j) denotes the quantity of labor employed by monopolist j, and A_t denotes the state of time-ttechnology, which evolves according to

log(A_t) =ρ_Alog(At−1) +ε_A,t, (30) where |ρ_A| < 1. The parameter ν > 1 controls the degree of substitutability between different intermediate inputs.

The foreign goods used in the production of the domestic final good (Y_F,t) and in the production of the foreign final good (Y_F,t^∗ ) are produced according to the technologies

YF,t= Z 1

0

XF,t(j)^ν−1ν dj _ν−1^ν

and Y_F,t^∗ = Z 1

0

X_F,t^∗ (j)^ν−1ν dj _ν−1^ν

. (31)

Here,XF,t(j) andX_F,t^∗ (j) are foreign intermediate goods produced by monopolistjusing the linear technology

X_F,t(j) +X_F,t^∗ (j) =A^∗_tL^∗_t(j) , (32) whereL^∗_t(j) is the labor employed by monopolistjin the foreign country andA^∗_t denotes the state of technology in the foreign country at timet, which evolves according to

log(A^∗_t) =ρ_Alog(A^∗_t−1) +ε^∗_A,t. (33)

Monopolists in the home country choose ˜PH,t(j) and ˜P_H,t^∗ (j) to maximize per-period profits given by

P˜_H,t(j)−W_t/A_t

X_H,t(j) +

N ER_tP˜_H,t^∗ (j)−W_t/A_t

X_H,t^∗ (j) , (34) subject to the demand curves of final good producers:

XH,t(j) =

P˜_H,t(j) PH,t

!−ν

YH,t and X_H,t^∗ (j) =

P˜_H,t^∗ (j) P_H,t^∗

!−ν

Y_H,t^∗ . (35) The aggregate price indexes forX_H,t and X_H,t^∗ , denoted by P_H,t and P_H,t^∗ , can be expressed as

PH,t≡ Z 1

0

hP˜H,t(j) i1−ν

dj _1−ν¹

and P_H,t^∗ ≡ Z 1

0

hP˜_H,t^∗ (j) i1−ν

dj _1−ν¹

. (36) Monopolists in the foreign country choose ˜P_F,t(j) and ˜P_F,t^∗ (j) to maximize profits given by

P˜_F,t^∗ (j)−W_t^∗/A^∗_t

X_F,t^∗ (j) +

N ER⁻¹_t P˜_F,t(j)−W_t^∗/A^∗_t

X_F,t(j) , (37) subject to the demand curves of final good producers:

XF,t(j) =

P˜F,t(j) P_F,t

!−ν

YF,t and X_F,t^∗ (j) =

P˜_F,t^∗ (j) P_F,t^∗

!−ν

Y_F,t^∗ . (38) Here, the aggregate price index forXF,tand X_F,t^∗ , denoted byPF,t andP_F,t^∗ , can be expressed as

P_F,t≡ Z 1

0

hP˜_F,t(j) i1−ν

dj _1−ν¹

and P_F,t^∗ ≡ Z 1

0

hP˜_F,t^∗ (j) i1−ν

dj _1−ν¹

. (39)

The first-order conditions for the monopolists imply

P˜_H,t(j) =N ER_tP˜_H,t^∗ (j) = ν ν−1

W_t

A_t, (40)

where ˜PH,t(j) and ˜P_H,t^∗ (j) are prices that the home monopolist charges in the home and foreign markets, respectively. Similarly,

N ER⁻¹_t P˜F,t(j) = ˜P_F,t^∗ (j) = ν ν−1

W_t^∗

A^∗_t . (41)

Here, ˜P_F,t(j) and ˜P_F,t^∗ (j) are the prices that the foreign monopolist charges in the home and foreign markets, respectively. Equations (40) and (41) imply that the law of one price holds for intermediate goods.

4.1.3 Monetary policy, market clearing, and the aggregate resource constraint In our first specification of monetary policy, the domestic monetary authority sets the nominal interest rate according to the following Taylor rule:

R_t= (Rt−1)^γ

Rπ^θ_t^π1−γ

exp (ε_R,t). (42)

We assume that the Taylor principle holds, so that θπ > 1. In addition, R = β⁻¹, and ε^R_t is an independently and identically distributed (iid) shock to monetary policy. To simplify, we assume that the inflation target is zero in both countries. The foreign monetary authority follows a similar rule:

R^∗_t = R^∗_t−1γ

R(π_t^∗)^θπ 1−γ

exp ε^∗_R,t

. (43)

We abstract from the output gap in the Taylor rule to ease the comparison between the flexible- price version of the model, which has a zero output gap, and the sticky-price version of the model.

In practice, the output gap coefficients in estimated versions of the Taylor rule are quite small (see, e.g. Clarida, Gali, and Gertler (1998)). Modifying the Taylor rule to include empirically plausible responses to the output gap has a negligible effect on our results.¹⁸

In our second specification of monetary policy, the domestic monetary authority sets the growth rate of the money supply according to:

log M_t

Mt−1

=x^M_t , where x^M_t =ρ_XMx^M_t−1+ε^M_t . (44) Here,|ρ_XM|<1 and ε^M_t is an iid shock to monetary policy. For convenience, we assume that the unconditional mean growth rate of nominal money balances is zero. The foreign monetary authority follows a similar rule so that

log M_t^∗

M_t−1^∗

=x^M_t ^∗, where x^M_t ^∗=ρXMx^M∗_t−1+ε^M∗_t . (45) In our third specification of monetary policy, the domestic monetary authority sets the nominal interest rate as in (42), but the foreign monetary authority uses an augmented Taylor rule that includes a term that targets the N ER.

R^∗_t =R

N ER^−θ_t ^{N ER}

exp ε^∗_R,t

. (46)

We assume that the Taylor principle holds so thatθπ >1, and thatθN ER>0 so that the nominal interest rate in the foreign country rises whenever there is a depreciation of the foreign currency.

We refer to the three specifications of monetary policy as the Taylor rule, the exogenous money growth rule, and the exchange rate targeting rule, respectively.

18Suppose we define the output gap as the percentage deviation of output from its steady-state value and include it in the Taylor rule with a coefficient equal to 0.5. The resulting impulse functions are very similar to those obtained for a version of the model with a Taylor rule that excludes the output gap.

We assume that government purchases,Gt, evolve according to:

log G_t

G

=ρ_Glog Gt−1

G

+ε^G_t , (47)

and that, without loss of generality, the government budget is balanced each period using lump- sum taxes. Here, |ρ_G| < 1 and ε^G_t is an iid shock to government purchases. The composition of government expenditures in terms of domestic and foreign intermediate goods (YH,tandYF,t) is the same as that of the domestic household’s final consumption good.

Similarly, government purchases in the foreign country,G^∗_t, evolve according to log

G^∗_t G

=ρGlog G^∗_t−1

G

+ε^G∗_t , (48)

where ε^G∗_t is an iid shock to government purchases and the government budget is balanced each period using lump-sum taxes. The composition of government expenditures in terms of domestic and foreign intermediate goods (Y_F,t^∗ and Y_H,t^∗ ) is the same as that of the foreign household’s final consumption good. Since bonds are in zero net supply, bond-market clearing implies

BH,t+B_H,t^∗ = 0 and BF,t+B_F,t^∗ = 0. (49) Labor market clearing requires that:

Lt= Z 1

0

Lt(j)dj and L^∗_t = Z 1

0

L^∗_t(j)dj. (50) Market clearing in the intermediate input markets requires that

X_H,t(j) +X_H,t^∗ (j) =A_tL_t and X_F,t(j) +X_F,t^∗ (j) =A^∗_tL^∗_t. (51) Finally, the aggregate resource constraints are given by

Yt=Ct+Gt and Y_t^∗ =C_t^∗+G^∗_t. (52) 4.1.4 Impulse response functions

In the examples in this section we use the following parameter values. We assume a Frisch elasticity of labor supply equal to 1 (φ = 1) and, as in Christiano, Eichenbaum, and Evans (2005), set σM = 10.62. We set the value of β so that the steady-state real interest rate is 3 percent. As in Backus, Kehoe, and Kydland (1992), we assume that the elasticity of substitution between domestic and foreign goods in the consumption aggregator is 1.5 (ρ = 1/3) and the import share is 10 percent (ω = 0.9) so that there is home bias in consumption. We set ν = 6, which implies an average markup of 20 percent. This value falls well within the range considered by Altig et al.

(2011). We normalize the value of χ, which affects the marginal disutility of labor, so that hours worked in the steady state is equal to 1. We assume that monetary policy is given by the Taylor